Flowfield in cylidrical coords

Homework Help
1

For an incompressible flow in the rθ plane, the r component of velocity is given as ur=U cos θ. (a) Determine a possible θ component of velocity. (b) How many possible θ components are there?

2

This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:

  1. ENGINEERING CONTEXT: In many fluid applications, such as flow around curved surfaces or through annular channels, a prescribed radial velocity in polar coordinates must be paired with a compatible circumferential (θ) velocity to satisfy incompressibility.

  2. PHYSICS: The dominant effect is mass conservation, which forces the two components of the velocity field to be coupled through the continuity equation in polar coordinates.

  3. ASSUMPTIONS: We assume a steady, two‐dimensional incompressible flow in the rθ plane with a smooth velocity field; these assumptions allow the use of the divergence-free condition without introducing additional complications.

  4. SOLUTION TECHNIQUE:
        a. The incompressible continuity equation in polar coordinates is

\frac{1}{r}\frac{\partial}{\partial r}\left(r u_r\right)+\frac{1}{r}\frac{\partial u_\theta}{\partial\theta}=0.

    b. Substituting the given radial velocity u_r=U\cos\theta yields

\frac{1}{r}\frac{\partial}{\partial r}\left(rU\cos\theta\right)+\frac{1}{r}\frac{\partial u_\theta}{\partial\theta}= \frac{U\cos\theta}{r}+\frac{1}{r}\frac{\partial u_\theta}{\partial\theta}=0.

    c. Multiplying through by r gives

U\cos\theta+\frac{\partial u_\theta}{\partial\theta}=0,

so that

\frac{\partial u_\theta}{\partial\theta}=-U\cos\theta.

    d. Integrating with respect to \theta results in

u_\theta=-U\sin\theta+f(r),

where f(r) is an arbitrary function of r arising from the integration.

  1. REFLECTION: The result shows that the circumferential velocity component is not unique; an arbitrary axisymmetric function f(r) may be added without violating mass conservation. In engineering applications, additional conditions (for example, boundary conditions and/or conservation of momentum) would uniquely determine the flow field.

To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
  2. Appendix F.3: Volumetric Viscous Force Part 1 (HIGHLY RELEVANT)
  3. Chapter 5.1: Inviscid Incompressible Flow (HIGHLY RELEVANT)
  4. Chapter 5 Engineering Models: Steady Euler Equations (HIGHLY RELEVANT)
  5. Chapter 3 Engineering Models: Hydrostatic Equation (HIGHLY RELEVANT)

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